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Open Journal of Statistics, 2014, 4, 597-610 Published Online
September 2014 in SciRes. http://www.scirp.org/journal/ojs
http://dx.doi.org/10.4236/ojs.2014.48056
How to cite this paper: Ghnimi, S. and Gasmi, S. (2014)
Parameter Estimations for Some Modifications of the Weibull
Dis-tribution. Open Journal of Statistics, 4, 597-610.
http://dx.doi.org/10.4236/ojs.2014.48056
Parameter Estimations for Some Modifications of the Weibull
Distribution Soumaya Ghnimi1, Soufiane Gasmi2 1Faculty of Sciences
of Tunis, University of Tunis El Manar, Tunis, Tunisia 2Tunis
National Higher School of Engineering, University of Tunis, Tunis,
Tunisia Email: [email protected] Received 20 May 2014;
revised 26 June 2014; accepted 12 July 2014
Copyright © 2014 by authors and Scientific Research Publishing
Inc. This work is licensed under the Creative Commons Attribution
International License (CC BY).
http://creativecommons.org/licenses/by/4.0/
Abstract Proposed by the Swedish engineer and mathematician
Ernst Hjalmar Waloddi Weibull (1887- 1979), the Weibull
distribution is a probability distribution that is widely used to
model lifetime data. Because of its flexibility, some modifications
of the Weibull distribution have been made from several researches
in order to best adjust the non-monotonic shapes. This paper gives
a study on the performance of two specific modifications of the
Weibull distribution which are the exponentiated Weibull
distribution and the additive Weibull distribution.
Keywords Exponentiated Weibull Distribution, Additive Weibull
Distribution, Maximum Likelihood Estimation, Kolmogorov-Smirnov
Test, Simulation
1. Introduction The Weibull distribution [1] is the most
life-time probability distribution used in the reliability
engineering dis-cipline. Due to its wide applications [2], many
researchers have developed various extensions and modified forms of
the Weibull distribution with a number of parameters ranging from 2
to 5. These distributions have several desirable properties and
nice physical interpretations. The literature that studies the
various modifica-tions of the Weibull distributions is extensive,
for example: the two-parameter flexible Weibull extension of
Bebbington et al. [3]. Zhang and Xie [4] studied the
characteristics and application of the truncated Weibull
dis-tribution which has a bathtub shaped hazard function.
A three-parameter model, called exponentiated Weibull
distribution, was introduced by Mudholkar and Sri-
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S. Ghnimi, S. Gasmi
598
vastave [5]. The modified Weibull distribution of Sarhan and
Zaindin [6] was studied by Gasmi and Berzig [7] in the case of type
I censored data. Another three-parameter model was developed by
Marshall and Olkin [8] and is called the extended Weibull
distribution. Xie et al. [9] proposed a three-parameter modified
Weibull ex-tension with a bathtub shaped hazard function. Lai et
al. [10] have described the modified Weibull (MW) dis-tribution. A
four-parameter additive Weibull distribution (AddW) was proposed by
Xie and Lai [11]. A second four-parameter beta Weibull distribution
was proposed by Famoye et al. [12]. Cordeiro et al. [13] introduced
another four-parameter distribution called the Kumaraswamy Weibull
distribution. A five-parameter modified Weibull distribution was
introduced by Phani [14]. The beta modified Weibull distribution
was introduced by Silva et al. [15] and further studied by Cordeiro
et al. [16]. Recently, an extensive review of some discrete and
continuous versions of the modifications of the Weibull
distribution was introduced by Almalki and Nadarajah [17]. The main
objective of this article is in first step to estimate the three
unknown parameters of the exponen-tiated Weibull distribution and
the four unknown parameters of the additive Weibull distribution.
Therefore, we use the maximum likelihood method to derive such
estimates. In the second step, we study whether these
distri-butions fit a set of real data of Aarset [18] better than
other distributions. Two criteria are used for this purpose: the
first one is the mean square distance MSD and the next one is the
Kolmogorov-Smirnov test statistic. A real data set is analyzed and
it is observed that the present distributions provide better fit
than many existing well-known distributions. This paper will be
organized as follows. In Section 2 we present the exponentiated
Weibull distribution and the additive Weibull distribution. In
Section 3, an application to real data is provided and different
types of goodness-of-fit are applied to test the compatibility of
the exponentiated Weibull distribu-tion and the additive Weibull
distribution in comparison to some other models. Mainly we use the
mean square distance MSD and the Kolmogorov-Smirnov (K-S) test as a
non-parametric test to illustrate how one can com-pare the
exponentiated Weibull distribution and the additive Weibull
distribution with some sub-models. Finally we conclude the paper in
Section 4.
2. Parameter Estimates of EW and AddW Distributions 2.1.
Exponentiated Weibull Distribution The exponentiated Weibull (EW)
distribution is proposed by Mudholkar and Srivastava [5] and
studied first by Mudholkar et al. [19] and further by Mudholkar and
Hutson [20].
The cumulative distribution function (CDF) and the survival
function of the EW distribution, denoted by ( )EW , ,α θ λ are
respectively:
( ) ( )( )EW ; 1 expF t t λθαΘ = − − , where ( ), ,α θ λΘ = and
, , 0α θ λ > (1) and
( ) ( )( )EW ; 1 1 expS t t λθαΘ = − − − (2) The ( )EW , ,α θ λ
distribution generalizes the following distributions: 1)
exponential distribution ( )ED α
by setting 1θ = , 1λ = , 2) Rayleigh distribution ( )RD α by
setting 2θ = , 1λ = , 3) generalized exponen-tial distribution (
)GED ,α λ [21] by setting 1θ = and 4) Weibull distribution ( )WD ,α
θ [22] [23] by set-ting 1λ = .
Figure 1 represents the cumulative distribution function and the
survival function of the ( )EW , ,α θ λ for different values of α ,
θ and λ .
The probability density function of the ( )EW , ,α θ λ
distribution is given by:
( ) ( ) ( )( ) 11; exp 1 exp , 0.f t t t t tλθ θ θλαθ α α −−Θ =
− − − > (3) The corresponding hazard function has the form:
( )( ) ( )( )
( )( )
11 exp 1 exp; .
1 1 exp
t t th t
t
λθ θ θ
θ
λαθ α α
α
−− − − −
Θ =− − −
(4)
Figure 2 shows respectively the probability density function and
the hazard rate function of the ( )EW , ,α θ λ
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S. Ghnimi, S. Gasmi
599
Figure 1. Plots of cumulative distribution function and survival
function of ( )EW , ,α θ λ .
Figure 2. Plots of probability density function and hazard rate
function of the ( )EW , ,α θ λ .
distribution for different values of α , θ and λ .
2.1.1. Data Simulations of the ( ), ,α θ λEW Distribution By
setting the three parameters α , θ and λ as follows: 1α = , 3θ =
and 1λ = , we obtain simulation data of a ( )EW , ,α θ λ
distribution. We remark that the ( )EW , ,α θ λ distribution has
the advantage that it pos-sessed a closed form of cdf, therefore we
can generate random values from it by using the explicit
formula:
( ) 11log 1i
Ut
θλ
α
− − =
,
where 1, ,i n= , n is the sample size and U is a uniformly
distributed random variable on the interval (0, 1).
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S. Ghnimi, S. Gasmi
600
Figure 3 illustrate the empirical cdf, the cdf and the 95% lower
and upper confidence bounds for the cdf of 100 simulated data by
setting 1α = , 3θ = and 1λ = .
2.1.2. Parameter Estimation To estimate the parameters of the (
)EW , ,α θ λ distribution we use the maximum-likelihood method
which is a traditional parametric method to estimate the parameters
and has good properties such as asymptotic normality and
consistency. Suppose now that we have a random sample, ( )1 2, , ,
nt t t of a ( )EW , ,α θ λ distribution with unknown parameter
vector ( ), , α θ λΘ = . The likelihood function of Θ is given
by:
( ) ( ) ( )( ) 111
; exp 1 exp .n
i i i ii
L t t t tλθ θ θλαθ α α−
−
=
Θ = − − −∏ (5)
The log-likelihood function has the following form:
( ) ( ) ( )( ) -111 1 11
; exp 1 exp .n
ii
L t t t tλθ θ θλαθ α α−
=
Θ = − − −∏ (6)
After calculating the first partial derivatives of ( )1ln ;L t Θ
and setting the results to zero, we get the follow-ing score
functions:
( )1 1
e0 11 e
i
i
tn ni
i ti i
tn tθ
θ
αθθ
αλ
α
−
−= =
= − + −−
∑ ∑ (7)
( ) ( ) ( ) ( )1 1 1
log e0 log log 1
1 e
i
i
tn n ni i
i i i ti i i
t tn t t tθ
θ
αθθ
αα α λ
θ
−
−= = =
= + − + −−
∑ ∑ ∑ (8)
( )1
0 log 1 e in
t
i
n θαλ
−
=
= + −∑ (9)
To get the MLE of the parameters α , θ and λ we have to solve
the above system of three non-linear eq-uations with respect to α ,
θ and λ . The solution of this system of equations is not possible
in closed form, so numerical technique such as the trust region
method, which requires the second derivatives of the ( )1ln ;L t
Θfunction is needed to get the MLE. We note that in order to
accelerate the resolution of the system (7), (8), (9) by using the
software MATLAB, we have introduced the following second partial
derivatives of ( )1ln ;L t Θ :
Figure 3. Cdf and empirical cdf of the ( )EW , ,α θ λ
distribution.
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S. Ghnimi, S. Gasmi
601
( )( )
22
2 2 21
eln 11 e
i
i
tni
ti
tL nθ
θ
αθ
αλ
α α
−
−=
∂= − − −
∂ −∑
( ) ( )( ) ( )
( )2
22
2 2 21 1
log e 1 eln log 11 e
i i
i
t tn n i i i
i iti i
t t tL n t t
θ θ
θ
α αθ θ
θ
α
αα α λ
θ θ
− −
−= =
− −∂= − − + −
∂ −∑ ∑
2
2 2ln L nλ λ
∂= −
∂
( ) ( )( ) ( )
( )2
21 1
log e 1 eln log 11 e
i i
i
t tn n i i i
i iti i
t t tL t t
θ θ
θ
α αθ θ
θ
α
αλ
α θ
− −
−= =
− −∂= − + −
∂ ∂ −∑ ∑
2
1
eln1 e
i
i
tni
ti
tLθ
θ
αθ
αα λ
−
−=
∂=
∂ ∂ −∑
( )21
log eln .1 e
i
i
tni i
ti
t tLθ
θ
αθ
αα
λ θ
−
−=
∂=
∂ ∂ −∑
Table 1 gives the estimated parameters of 10N = simulations and
the mean square error of each parameter, where:
( ) ( )21
1ˆ ˆMSEN
iiN =
Θ = Θ−Θ∑
2.2. Additive Weibull Distribution The additive Weibull (AddW)
distribution has four parameters α , β , θ and γ . This
distribution is first in-troduced by Xie and Lai [11] and is
denoted by ( )AddW , , ,α β θ γ . We remark, that this distribution
has a bathtub shaped hazard function and it was obtained as the sum
of two hazard functions of Weibull distributions.
The cumulative distribution function of the ( )AddW , , ,α β θ γ
is defined as follows:
Table 1. Parameter estimates of 1α = , 3θ = and 1λ = .
ML
α̂ θ̂ λ̂
1.1002 2.9859 1.0415
1.0217 3.0602 1.0954
1.0930 3.0329 0.9756
1.0293 3.0174 1.0183
1.0724 3.0286 0.9842
1.0903 3.0802 0.9988
0.9929 3.0472 0.9904
1.0081 2.9723 1.0480
0.9905 2.9729 0.9863
0.9897 3.0126 1.0208
MSE 0.0026 0.0016 0.0015
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602
( ) ( )AddW ; 1 exp ,F t t tθ γα βΘ = − − − where ( ), , , ,α β
θ γΘ = , , 0α β θ > and 1.γ < (10) The corresponding survival
function is:
( ) ( )AddW ; exp .S t t tθ γα βΘ = − − (11) The ( )AddW , , ,α
β θ γ distribution generalizes the following distributions: 1)
linear failure rate distribution
( )LRFD ,α β [7] by setting 2γ = and 1θ = , 2) Weibull
distribution ( )WD ,α θ by setting 0β = and 3) modified Weibull
distribution ( )MWD , ,α β γ [10] by setting 1θ = .
Figure 4 shows respectively the cumulative distribution function
and the survival function of the additive Weibull distribution for
different values of α , β , θ and γ .
The probability density function of the ( )AddW , , ,α β θ γ
distribution is given by: ( ) ( ) ( )1 1; exp , 0f t t t t t tθ γ θ
γα β α β− −Θ = − − − − > (12)
The corresponding hazard function has the form:
( ) 1 1;h t t tθ γαθ βγ− −Θ = + (13) Figure 5 shows the
probability density function and the hazard rate function of the (
)AddW , , ,α β θ γ dis-
tribution for different values of α , β , θ and γ .
2.2.1. Data Simulations of the ( )AddW , , ,α β θ γ Distribution
By setting the four parameters α , β , θ and γ as follows: 1.5α = ,
0.5β = , 3θ = and 0.8γ = , we obtain simulation data of the ( )AddW
, , ,α β θ γ distribution. We generate random values from it by
solving the following equation:
( )log 1 0,i iU t tθ γα β− + + = where 1, ,i n= , n is the
sample size and U is a uniformly distributed random variable on the
interval (0, 1).
Figure 6 illustrates the empirical cdf, the cdf and the 95%
lower and upper confidence bounds for the cdf of the 100 simulated
data by setting 1.5α = , 0.5β = , 3θ = and 0.8γ = .
2.2.2. Parameter Estimation Now, we introduce the estimation of
the model parameters by using the method of maximum likelihood.
Let
Figure 4. Plots of cumulative distribution function and survival
function of the ( )AddW , , ,α β θ γ .
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603
Figure 5. Plots of probability density function and hazard rate
function of the ( )AddW , , ,α β θ γ .
Figure 6. Cdf and empirical cdf of the ( )AddW , , ,α β θ γ
.
( )1 2, , , nt t t be a random sample of the AddW distribution
with unknown parameters α , β , θ and γ . By setting ( ), , ,α β θ
γΘ = , the likelihood function of this sample is given by:
( ) ( ) ( )1 11
; exp .n
i i i i ii
L t t t t tθ γ θ γαθ βγ α β− −=
Θ = + − −∏ (14)
The log-likelihood function has the following form:
( ) ( ) ( )1 11
ln ; log .n
i i i i ii
L t t t t tθ γ θ γαθ βγ α β− −=
Θ = + − +∑ (15)
After calculating the first partial derivatives of ( )ln ;iL t Θ
and setting the obtained expressions equal to zero, we get the
following score functions:
1
1 11 1
0n n
ii
i ii i
t tt t
θθ
θ γ
θαθ βγ
−
− −= =
= −+∑ ∑ (16)
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604
1
1 11 1
0n n
ii
i ii i
t tt t
γγ
θ γ
γαθ βγ
−
− −= =
= −+∑ ∑ (17)
( ) ( )1 1
1 11 1
log0 log
n ni i i
i ii ii i
t t tt t
t t
θ θθ
θ γ
α αθα
αθ βγ
− −
− −= =
+= −
+∑ ∑ (18)
( ) ( )1 1
1 11 1
log0 log
n ni i i
i ii ii i
t t tt t
t t
γ γγ
θ γ
β βγβ
αθ βγ
− −
− −= =
+= −
+∑ ∑ (19)
To get out the MLE of the unknown parameters, we have to solve
the above system of four non-linear equa-tions with respect to α ,
β , θ and γ . The solution of this system of equations is not
possible in closed form, so numerical technique such as the trust
region method is needed to get the MLE.
We obtain the second partial derivatives of ( )ln ;iL t Θ as
follows:
( )( )
212
2 21 11
ln n ii
i i
tL
t t
θ
θ γ
θ
α αθ βγ
−
− −=
−∂=
∂ +∑
( )( )
212
2 21 11
ln n ii
i i
tL
t t
γ
θ γ
γ
β αθ βγ
−
− −=
−∂=
∂ +∑
( ) ( ) ( )( )
( )21 1 2 1 1 12
22 21 11 1
2 log logln logn ni i i i i i i
i ii i
i i
t t t t t t tL t tt t
θ γ θ γ θθ
θ γ
αβγ αβθγ αα
θ αθ βγ
− − − − −
− −= =
+ −∂= −
∂ +∑ ∑
( ) ( ) ( )( )
( )21 1 2 1 1 -12
22 21 11 1
2 log logln logn ni i i i i i i
i ii i
i i
t t t t t t tL t tt t
θ γ θ γ γγ
θ γ
αβθ αβθγ ββ
γ αθ βγ
− − − −
− −= =
+ −∂= −
∂ +∑ ∑
( )1 12
21 11
ln n i ii
i i
t tL
t t
θ γ
θ γ
θγα β αθ βγ
− −
− −=
−∂=
∂ ∂ +∑
( )( )
( )1 1 1 12
21 11 1
logln logn n
i i i i ii i
i ii i
t t t t tL t tt t
θ γ θ γθ
θ γ
βγ βθγα θ αθ βγ
− − − −
− −= =
+∂= −
∂ ∂ +∑ ∑
( )( )1 1 1 12
21 11
logln n i i i i ii
i i
t t t t tL
t t
θ γ θ γ
θ γ
βθ βθγα γ αθ βγ
− − − −
− −=
− −∂=
∂ ∂ +∑
( )( )1 1 1 12
21 11
logln n i i i i ii
i i
t t t t tL
t t
θ γ θ γ
θ γ
αγ αθγβ θ αθ βγ
− − − −
− −=
− −∂=
∂ ∂ +∑
( )( )
( )1 1 1 12
21 11 1
logln logn n
i i i i ii i
i ii i
t t t t tL t tt t
θ γ θ γγ
θ γ
αθ αθγβ γ αθ βγ
− − − −
− −= =
+∂= −
∂ ∂ +∑ ∑
( )( ) ( )( )( )
1 1 1 12
21 11
log logln n i i i i i ii
i i
t t t t t tL
t t
θ θ γ γ
θ γ
α αθ β βγ
θ γ αθ βγ
− − − −
− −=
− + −∂=
∂ ∂ +∑
Table 2 gives the estimated parameters of 10 simulations and the
mean square error of each parameter.
3. Analysis of a Real Data Set In this section, we analyze a
real data set to demonstrate the performance of the EW and AddW
distributions in
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S. Ghnimi, S. Gasmi
605
practice. A sample of 50 components taken from Aarset [18] has
been studied. For this data set, we compare at first the results of
the fits of the EW distribution (EWD) against ED, GED, RD and WD
which are sub-models of the EW distribution. In the second step the
fits of the AddW distribution (AddWD) will be compared against WD,
MWD, and LRFD which are sub-models of the AddW distribution. Table
3 gives the often used lifetimes of 50 devices introduced by
Aarset. Table 4 and Table 5 show the MLE of the parameters, the
log-likelihood function values and the MSD on the one hand for the
ED, RD, GED, WD, EWD and on the other hand for the WD, MWD, LRFD,
and the AddW models. Table 6 and Table 7 show the observed K-S test
statistic values for each models EWD and AddWD and their
correspondent sub-models and the p-value for each one. Figure 7 and
Figure 8 show the plots of the empirical and fitted scaled
TTT-Transforms, the empirical and parametric cumu-lative density
functions, the empirical and fitted hazard and probability density
functions for the models EWD, AddWD and their correspondent
submodels.
However in Figure 9 we have a comparison between the two models
EW and AddW. We note that for com-parison purpose, we use the mean
square difference between the empirical cdf and the fitted cdf,
denoted by MSD. The MSD is computed by the following relation:
( )21
1 ˆMSDr
i ii
F Fr =
= −∑ ,
where îF and iF are the estimated and the empirical cdf
computed at the cumulative failure times it and r is the size of
the data set.
Based on the results shown in Table 4 and Table 5, we could
deduce that: • compared with the MSD of the ED and the WD, the EWD
is not the best fit of the Aarset data; • the MSD of the AddWD has
the lowest value compared with each sub-models, so the AddWD is the
best in
fitting the Aarset data; • the MSD of the AddWD is smaller than
the MSD of the EWD which indicates that the AddWD fits the
given
data better than the EWD.
Table 2. Parameter estimates of 1.5α = , 0.5β = , 3θ = and 0.8γ
= .
ML
α̂ β̂ θ̂ γ̂
1.5457 0.5406 3.0174 0.8607
1.5305 0.5294 3.0610 0.8008
1.5989 0.5090 2.9544 0.8543
1.5923 0.4939 2.9261 0.8389
1.4958 0.4909 3.0615 0.8300
1.4336 0.5443 3.0996 0.8297
1.4493 0.5754 3.0863 0.8433
1.4821 0.5702 3.1001 0.8461
1.4579 0.5419 2.9718 0.8142
1.4308 0.4620 3.0760 0.8101
MSE 0.0034 0.0018 0.0050 0.0014
Table 3. Lifetimes of 50 devices, Aarset.
0.1 0.2 1 1 1 1 1 2 3 6 7 11 12 18 18 18 18 18
21 32 36 40 45 46 47 50 55 60 63 63 67 67 67 67 72 75
79 82 82 83 84 84 84 85 85 85 85 85 86 86
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606
Table 4. MLE of the parameter(s), log-likelihood function values
and the MSD of sub-mod- els of the EWD.
The Model Parameter estimates ln L MSD
RD 4ˆ 3.1809 10α −= × −264.052 0.0153
GED ˆ 0.01870α = , ˆ 0.7798λ = −239.995 0.0152
EWD 10ˆ 6.4800 10α −= × , ˆ 4.6900θ = , ˆ 0.1460λ = −229.114
0.0151
WD ˆ 0.0270α = , ˆ 0.9490θ = −241.002 0.0139
ED ˆ 0.0219α = −241.089 0.0136
Table 5. MLE of the parameter(s), log-likelihood function values
and the MSD of sub-mod- els of the AddWD.
The model Parameter estimates ln L MSD
LRFD ˆ 0.0140α = , 4ˆ 2.4000 10β −= × −238.064 0.0282
WD ˆ 0.0270α = , ˆ 0.9490θ = −241.002 0.0139
MWD ˆ 0.0120α = , 8ˆ 2.1590 10β −= × , ˆ 4.0140γ = −230.510
0.0072
AddWD 5ˆ 3.9345 10α −= × , ˆ 0.0860β = , ˆ 2.3760θ = , ˆ 0.4114γ
= −228.102 0.0058
Table 6. The MLE of the parameter(s), K-S values and the
associated p-values.
The model Parameter estimates K-S p-value
RD 4ˆ 3.1809 10α −= × 0.2552 0.0328
GED ˆ 0.0187α = , ˆ 0.7798λ = 0.1775 0.2678
WD ˆ 0.0270α = , ˆ 0.9490θ = 0.1657 0.3440
ED ˆ 0.0219α = 0.1601 0.3846
EWD 10ˆ 6.48 10α −= × , ˆ 4.69θ = , ˆ 0.146λ = 0.1490 0.4732
Table 7. The MLE of the parameter(s), K-S values and the
associated p-values.
The model Parameter estimates K-S p-value
LRFD ˆ 0.0140α = , 4ˆ 2.4000 10β −= × 0.2057 0.1366
WD ˆ 0.0270α = , ˆ 0.9490θ = 0.1657 0.3440
MWD ˆ 0.0120α = , 8ˆ 2.159 10β −= × , ˆ 4.0140λ = 0.1655
0.3453
AddWD 5ˆ 3.9345 10α −= × , ˆ 0.0860β = , ˆ 2.3760θ = , ˆ 0.4114λ
= 0.1230 0.7089
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S. Ghnimi, S. Gasmi
607
Figure 7. (a) The empirical and estimated scaled TTT-Transform
plots of the ED, RD, GED, WD and EWD models; (b) The empirical and
estimated cumulative density function of the ED, RD, GED, WD and
EWD models; (c) Empirical and estimated hazard rate functions of
the ED, RD, GED, WD and EWD models; (d) Empirical and estimated PDF
of the ED, RD, GED, WD and EWD models, for Aarset data.
Figure 8. (a) The empirical and estimated scaled TTT-Transform
plots of the WD, LRFD, MWD and AddWD models; (b) The empirical and
estimated cumulative density function of the WD, LRFD, MWD and
AddWD models; (c) Empirical and estimated hazard rate functions of
the WD, LRFD, MWD and AddWD models; (d) Empirical and estimated pdf
of the WD, LRFD, MWD and AddWD models, for Aarset data.
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S. Ghnimi, S. Gasmi
608
Figure 9. (a) The empirical and estimated scaled TTT-Transform
plots of the EWD and AddWD models; (b) The empirical and estimated
cumulative density function of the EWD and AddWD models; (c)
Empirical and estimated hazard rate functions of the EWD and AddWD
models; (d) Empirical and estimated PDF of the EWD and AddWD
models, for Aarset data.
We perform in the next step at first the test of the following
null hypotheses: 1) H0: 1α = , 1λ = , the data follow ED
“Exponential distribution”, 2) H0: 2θ = , 1λ = , the data follow RD
“Rayleigh distribution”, 3) H0: 1θ = , the data follow GED
“Generalized exponential distribution”, 4) H0: 1λ = , the data
follow WD “Weibull distribution”,
in favor of the alternative hypothesis Ha: the data follow the
EWD “Exponentiated Weibull distribution”. And on the other hand the
test of the following null hypotheses. 1) H0: 0β = , the data
follow WD “Weibull distribution”, 2) H0: 1θ = , 0γ > , the data
follow MWD “Modified Weibull distribution”, 3) H0: 1θ = , 2γ = ,
the data follow LRFD “Linear failure rate distribution”,
in favor of the alternative hypothesis Ha: the data follow the
AddWD “Additive Weibull distribution”. In the following, we use a
non parametric test statistics, Kolmogorov-Smirnov (K-S) test with
a level of signi-
ficance equal to 0.05, to test the null hypothesis mentioned
below against Ha. We accept H0 with the p-value under the condition
p-value > 0.05.
If we compare the EWD model with the sub-models ED, RD, GED and
WD, we can conclude from Table 6 that: • only the RD is rejected at
level 0.033ν ≥ ; • all H0’s excepted the RD are not rejected at
0.26ν ≤ ; • the EWD is the best model among those discussed here,
to fit the current data set because it has the biggest
p-value (0.4732) and the lowest K-S value (0.1490). Similarly,
when we compare the AddWD model with the sub-models WD, LRFD and
MWD, we can con-
clude from Table 7 that: • none of H0’s is rejected at level
0.13ν ≤ ; • the AddWD is the best model among those discussed here,
to fit the current data set because it has the big-
gest p-value (0.7089) and the lowest K-S value (0.1230); • the
AddWD is the best model among the EWD model to fit the current data
set because it has the lowest K-S
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S. Ghnimi, S. Gasmi
609
value (0.1230). We can immediately observe from Figure 7, Figure
8 and Figure 9 that: 1) the data set has a bathtub shaped
hazard rate, 2) one can see the closeness of the fitted pdf
using the AddWD model, 3) the AddWD fits the data set better than
all other distributions used here, because its fitted curve is
closer to the empirical curve.
4. Conclusion In this paper, we show the performance of two
models called the exponentiated Weibull distribution and the
ad-ditive Weibull distribution by using an empirical comparison
with the sub-models of each one such as the expo-nential
distribution, the Rayleigh distribution, the generalized Weibull
distribution, the linear failure rate distri-bution, the Weibull
distribution and the modified Weibull distribution. The maximum
likelihood estimations of the unknown parameters for these
distributions are discussed. A real data set of Aarset is studied
by using the EW and the AddW distributions. The results of the
comparisons showed that the additive Weibull distribution provided
a better fit for the Aarset data set than some of the often-used
distributions.
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Parameter Estimations for Some Modifications of the Weibull
DistributionAbstractKeywords1. Introduction2. Parameter Estimates
of EW and AddW Distributions2.1. Exponentiated Weibull
Distribution2.1.1. Data Simulations of the Distribution2.1.2.
Parameter Estimation
2.2. Additive Weibull Distribution2.2.1. Data Simulations of the
Distribution2.2.2. Parameter Estimation
3. Analysis of a Real Data Set4. ConclusionReferences